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Since, in this discussion, my name has been associated with a fix taken
with four bodies at the cardinal points, I have a few observations to make.


George wrote:

> > ... it should be noted that whenever such a claim is made, that the
> > true position is three times more likely to be outside rather than inside a
> > cocked hat, it is (or should be) noted that any systematic errors have been
> > first corrected for.

Peter replied:


>Sorry George, but this seems to be quite illogical nonsense.  You
>don't seem to have grasped the basic idea, and have thus tried to put
>the cart before the horse.  You CAN'T 'correct systematic errors
>first' since their correction leads to a single point - a fix
>position.  How are you then going to go about correcting for your ...
>whatever you like to call the other, erratic variety?
>
>This is WHY erratic error gets corrected at source, leaving an
>assumption that what remains must be ... constant, or systematic, or
>whatever you feel happy about naming it.

My response:

Why can't you correct for systematic error first? The point about
systematic errors is that for any particular instrument, they can be
identified and measured and thus accounted for in a measurement on that
instrument.

I have used this example before, but as it seems appropriate I make no
apologies in using it again.

Look at http://www.pisces-press.com/C-Nav/nav-plots/14%20March%20big.jpg
and you will see a plot of an actual fix in the Egyptian Western desert,
worked up with observations of bodies at (or close to) the cardinal points.
There are a number of points to note:

First, each of those LOPs is actually the result of five measurements on
the body itself. Averages were taken of the five altitudes and times of
observation - usually a minute apart - and then an Hc and Zn was calculated
for the average time and altitude. Peter's method is to plot altitude
against time and draw a straight line through the results. This amounts to
much the same thing and - if the straight line is drawn correctly - my
average should sit on a straight line fit through the results. Of course,
there are statistically kosher ways to make straight line fits to data, but
an eyeball fit actually works very well.

As George has mentioned, taking averages of data to minimize random error
in data does not get rid of the random error, as Peter seems to imply. It
only reduces the random error to acceptable limits. (If the random error is
still unacceptable, take more data points!)

I would also caution against throwing out data. In my experience, I have
ended up with a better Hc by keeping all the altitudes - even the "weirdo"
ones. It is fairly easy to see if a succession of sights on the same body
are giving good results. Given that the time between sights will be fairly
constant, the difference in measured altitude should also be fairly
constant. If I see it is wandering about for some reason, I take more
sights and let statistics take care of whatever is getting in the way of me
taking accurate sights.

Second, it will be seen that the LOPs in the picture form a box, with the
bodies observed always on the far side of the box. From this, I can deduce
that the dominant error in the data is now a systematic error, which in
this case is an index error due to the desert heat warping the sextant or
index arm slightly.

Given that the box is about 10 minutes of arc on a side, I can see that
there is some 5 minutes of  unaccounted for systematic error in the data.

Now look at
http://www.pisces-press.com/C-Nav/nav-plots/19%20March%20big.jpg for a fix
taken a few days later, at our next campsite a little further North. This
time I included the extra 5 minutes of systematic error which I found a few
days earlier and the result is pretty much the single point fix Peter was
talking about.

So, to sum up, this is an example where I DID "correct systematic errors
first" and where their correction did lead to a "single point fix". (Of
course, where the pen is drawing a line which is about a minute wide, a
"single point fix" should be taken in context.)

It should be noted though, that the final fix was still a minute or so away
from the actual position. In this case, it is not possible to determine if
the residual errors are random or systematic. There are ALWAYS residual
errors in data. Where random and systematic errors have been reduced to the
precision of the instrument, it is not worth the effort to try and reduce
these errors still further. With this instrument (an A-12 bubble sextant)
an accuracy of 1 or 2 nautical miles on the fix is as good as it is going
to get.

On the subject of "COCKED HATS".

In the past, I have always agreed with George that where there is no
systematic error in the measured altitudes, the probability of the actual
position being outside the cocked hat is three times higher than being
inside. But my position has shifted slightly on this one. I wonder if I can
induce George to come along with me. (In the following analysis, assume
that there is no systematic error of any kind).

First, consider a measured altitude on Polaris. On reducing the sight, I
will end up with an LOP which will run East-West across the chart. What can
I say about my actual position relative to this line? I can say that the
probability of my position being on or near one point of the line is equal
to that for any other point on the line. (In other words, I have no
information regarding my longitude). I can say that the probability of my
actual position being North of the line is 0.5, exactly the same as the
probability of being South of the line.

Now, suppose I take a sight on the star Canopus when it is on the meridian
to the South. I reduce the sight and draw another LOP on the chart. This
LOP also runs East-West across the chart, but I find that it is about 2
nautical miles (say) to the South of the Polaris LOP.

What can I say about my position relative to this new Canopus LOP? Since it
runs parallel to the Polaris line, I still have no new information about my
longitude. The chance of me being on or near one point of this line is just
the same as for any other point. What can I say about my position North or
South of the line? Is the probability of my actual position being to the
North of the Canopus LOP still 0.5, exactly the same as it being South of
the line? I would now say no.

Because the Polaris LOP sits to the North of the Canopus LOP, I have an
independent piece of data which indicates that my position is actually more
likely to be North of the Canopus LOP than South of it. Statistically
speaking, this is a so-called "Bayesian" approach to statistical analysis.
I have some a priori information regarding my actual position, which can
inform my estimation of my position with respect to the Canopus LOP.

Actually, there is no reason why I should have taken the Polaris sighting
first. I could just have easily have taken the Canopus sighting first. That
being the case, I can say by symmetry that the Canopus LOP can similarly
inform, regarding my position relative to the Polaris LOP. Now, I can say
that the probability of my position being South of the Polaris LOP is
higher than the probability of being North of it.

I will stop there as I am sure George will see where I am headed with this
argument when applied to the "cocked hat" problem. And since the essence of
the argument is already laid out, I need go no further to persuade George
if he is willing to come with me this far.

Geoffrey Kolbe


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